# Linear independence and Basis vectors

### Linearly dependent

A set of vectors is set to be** linearly dependent** if at least one of the vectors in the set can be defined as a linear combination of the others.

**Linearly dependent** vectors may either overlap with each other or define the same plane. Such extra vector is not necessary to define a plane or space. Below is a relation between vectors **u, v and w **and scalars.

**u** = *a***v** + *b***w (3D space)**
**w** = *a***v (2D space)**

### Linearly independent

If no vector in the set can be written as a linear combination of the others, the vectors are said to be **linearly independent**. Below is a relation between vectors **u, v and w **and scalars.

**u** != *a***v** + *b***w (3D space)**
**w** != *a***v (2D space)**

Two vectors **u** and **v** are **linearly independent** if the only numbers *a* and *b* satisfying *a***u**+*b***v **= 0 are [a = b = 0] (known as trivial solution)

## Basis vectors

**Basis vectors** are **lineary independent **vectors having the property that every vector in the space can be written uniquely as a linear combination of them. In other words, we can be scale basis vectors using scalars to reach all possible vectors in the given space.

Link: MIT Courseware 3Blue1Brown

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